<p> Lecture Notes for</p><p>Statistical Methods for Business I BMGT 211</p><p>Chapters 5 and 6</p><p>Professor Ahmadi, Ph.D.</p><p>Department of Management</p><p>Revised May1, 2004 Professor Ahmadi’s Lecture Notes Page 2 Chapter 5 Formulas</p><p>Required Conditions for a Discrete Probability Function</p><p> f(x) > 0</p><p> f(x) = 1</p><p>Discrete Uniform Probability Function</p><p> f(x) = 1/n where n = the number of values the random variable may assume</p><p>Expected Value of a Discrete Random Variable</p><p>E(x) = µ =  (x f(x))</p><p>Variance of a Discrete Random Variable</p><p>2  2 Variance (x) =  = (x - µ) f(x</p><p>Number of Experimental Outcomes Providing Exactly x Successes in n Trials</p><p> n n!   =  x x!(n - x)! where n! = n (n - 1) (n - 2) . . . (2)(1) (Remember: 0! = 1)</p><p>Binomial Probability Function</p><p> n! f(x) = p x (1 - p) n – x where x = 0 ,1, 2, ..., n x!(n - x)!</p><p>The Mean of a Binomial Distribution</p><p>µ = n p</p><p>The Variance of a Binomial Distribution</p><p>2 = n p (1 - p)</p><p>Professor Ahmadi’s Lecture Notes Page 3 Chapter 5</p><p>Discrete Probability Distributions</p><p>The manager of the university bookstore has kept records of the number of diskettes sold per day. She provided the following information regarding diskettes sales for a period of 60 days:</p><p>Number of Number Diskettes Sold of Days 0 6 1 9 2 12 3 18 4 12 5 3 a. Identify the random variable b. Is the random variable discrete or continuous? c. Develop a probability distribution for the above data. d. Is the above a proper probability distribution?</p><p> e. Develop a cumulative probability distribution.</p><p> f. Determine the expected number of daily sales of diskettes.</p><p> g. Determine the variance and the standard deviation.</p><p> h. If each diskette yields a net profit of 50 cents, what are the expected yearly profits from the sales of diskettes?</p><p>Professor Ahmadi’s Lecture Notes Page 4 Chapter 5</p><p>Introduction to Binomial Distribution</p><p>A production process has been producing 10% defective items. A random sample of four items is selected from the production process. a. What is the probability that the first 3 selected items are non-defective and the last item is defective?</p><p> b. If a sample of 4 items is selected, how many outcomes contain exactly 3 non-defective items?</p><p> c. What is the probability that a random sample of 4 contains exactly 3 non-defective items?</p><p> d. Determine the probability distribution for the number of non-defective items in a sample of four.</p><p> e. Determine the expected number (mean) of non-defectives in a sample of four.</p><p> f. Find the standard deviation for the number of non-defectives.</p><p>Professor Ahmadi’s Lecture Notes Page 5 Chapter 5</p><p>POISSON PROBABILITY DISTRIBUTION</p><p>During the registration period, students consult their advisor for course selection. A particular advisor noted that during each half hour an average of eight students came to see him for advising. a. What is the probability that during a half hour period exactly four students will consult him?</p><p> b. What is the probability that during a half hour period less than three students will consult him?</p><p> c. What is the probability that during an hour period ten students will consult him?</p><p> d. What is the probability that during an hour and fifteen minute period thirty students will consult him?</p><p>Professor Ahmadi’s Lecture Notes Page 6 Chapter 6 Formulas</p><p>Uniform Probability Density Function for a Random Variable x:</p><p> 1  for a  x  b b - a f(x) =  0 elsewhere </p><p>Mean and Variance of a Uniform Continuous Probability Distribution:</p><p> a + b (b - a)2  = 2  2 12</p><p>The Z Transformation Formula:</p><p>(x - ) z = </p><p>Solving for x using the Z transformation formula: x    Z</p><p>Professor Ahmadi’s Lecture Notes Page 7 Chapter 6</p><p>Continuous Probability Distributions</p><p>I. - The Uniform Distribution</p><p>The driving time for an individual from her home to her work is uniformly distributed between 300 to 480 seconds. a. Give a mathematical expression for the probability density function.</p><p> b. Compute the probability that the driving time will be less than or equal to 435 seconds.</p><p> c. Determine the probability that the driving time will be exactly 400 seconds.</p><p> d. Determine the expected driving time. e. Determine the standard deviation of the driving time.</p><p>Professor Ahmadi’s Lecture Notes Page 8 Chapter 6</p><p>II. - The Normal Distribution</p><p>1. Given that Z is the standard normal random variable, give the probabilities associated with the following:</p><p> a. P(Z < - 2.09) = ?</p><p> b. P(Z > -0.95) = ?</p><p> c. P(-2.55 < Z < -2.33) = ?</p><p>2. Z is a standard normal variable. Find the value of Z in the following:</p><p> a. The area between -Z and zero is 0.4929. Z = ?</p><p> b. The area to the right of Z is 0.0192.Z = ?</p><p> c. The area between -Z and Z is 0.668. Z = ?</p><p>Professor Ahmadi’s Lecture Notes Page 9 3. The weight of certain items produced is normally distributed with a mean weight of 60 ounces and a standard deviation of 8 ounces.</p><p> a. What percentage of the items will weigh between 50.4 and 72 ounces?</p><p> b. What percentage of the items will weigh between 42 and 52 ounces?</p><p> c. What percentage of the items will weigh at least 74.4 ounces?</p><p> d. What are the minimum and the maximum weights of the middle 60% of the items?</p><p>Professor Ahmadi’s Lecture Notes Page 10 4. Sun Love grapefruit growers have determined that the diameter of their grapefruits is normally distributed with a mean of 4.5 inches and a standard deviation of 0.3 inches. (You can find the step-by-step solution to this problem in my workbook.)</p><p> a. What is the probability that a randomly selected grapefruit will have a diameter of at least 4.14 inches?</p><p> b. What percentage of grapefruits has a diameter between 4.8 to 5.04 inches?</p><p> c Sun Love packs their largest grapefruits in a special package called "Super Pack." If 5% of all their grapefruits are packed in "Super Packs," what is the smallest diameter of the grapefruits, which are in the "Super Packs?"</p><p> d In this year's harvest, there were 111,500 grapefruits, which had a diameter over 5.01 inches. How many grapefruits has Sun Love harvested this year?</p><p>Professor Ahmadi’s Lecture Notes Page 11 5. In grading eggs, 30% are marked small, 45% are marked medium, 15% are marked large, and the rest are marked extra-large. If the average weight of the eggs is normally distributed with a mean of 3.2 ounces and a standard deviation of 0.6 ounces:</p><p> a What are the smallest and the largest weights of the medium size eggs?</p><p> b What is the weight of the smallest egg, which will be in the extra-large category?</p><p>Professor Ahmadi’s Lecture Notes Page 12</p>